1. Field of the Invention
The present invention relates to control of nonlinear systems and, more particularly, to a control system for tracking, i.e., sustaining, chaotic transients in a nonlinear, dynamic system, whereby tracking is meant that a parameter is varied.
2. Background of the Invention
Classical control techniques utilize instantaneous feedback or feedback based on sampling. Such techniques are generally based on linear or small signal modeling. However, in the absence of robust, accurate, analytical models, it is difficult to control nonlinear systems with such techniques. Some nonlinear systems, such as multimode lasers with nonlinear crystals, are not amenable to accurate modeling. Classical control of such systems may be performed, if at all, on a trial and error basis and may be extremely difficult, expensive and inflexible. When control in such systems is lost, very little is done except to recognize such loss of control. For a discussion of such control systems see, e.g., U.S. Pat. No. 5,163,063 to Yoshikawa et al.
Chaos can be a desirable feature in many applications. In biology, the disappearance of chaos may signal pathological phenomena. In mechanics, chaos could be induced in order to prevent resonance, such as in a system of coupled pendulums, where one can excite chaotic motion of several modes spreading the energy over a wide frequency range. In optics, material damage is caused by lasers having a peak intensity at a given temporal frequency, so chaos is desirable since it has broadband spectra. It has also been suggested that chaos occurs for normal machine tool cutting, making chaos preservation a desired control for deeper than normal cutting.
Recent developments in nonlinear dynamics have shown that most nonlinear systems have steady-state, periodic or chaotic attractors in phase space. The chaotic attractors contain an infinite number of unstable periodic orbits. A technique (the xe2x80x9cOGY techniquexe2x80x9d) has been reported for stabilizing the nonlinear system in the neighborhood of unstable orbits by directing subsequent iterates towards the local stable manifold of the selected orbit as described by Ott, Grebogi and Yorke in xe2x80x9cControlling Chaos,xe2x80x9d Phys. Rev. Left. 64 1196, (1190). Upon application of the OGY technique, the nonlinear system remains on that particular orbit.
A disadvantage of the OGY technique is that its application is not feasible if the periodic orbit or unstable steady state near which control is desired is not known. Furthermore, some operating conditions which are not in the neighborhood of any identifiable orbit will not be accessible to control. In addition, the OGY technique is not amenable to tracking the system over a wide range of operating conditions since the control becomes less effective as the operating point is brought further away from the orbit or state.
An additional problem of maintaining chaos occurs when there exists a chaotic transient in the presence (e.g., neighborhood) of a periodic attractor. One method of sustaining chaotic transients in the presence of another non-chaotic attractor includes using the natural dynamics of unstable states laying on the basin boundary separating a periodic attractor from chaotic transients, referred to as basin saddles. Although there is technically only one attractor since chaos is a transient, there is still a stable manifold which separates the chaotic transient from the periodic attractor. Once the flow gets in a neighborhood of a basin saddle, small perturbations of an accessible system parameter are used to redirect the flow towards the chaotic transient region. This is done by a targeting technique which uses the linearization of the flow about the saddle.
These and other current approaches to sustaining chaos use an algorithm that requires accurate knowledge of the dynamics in the region where chaos disappears, as well as knowledge of the pre-iterates of this region. Typically, these approaches employ an analytic scheme for sustaining chaos that uses state variable control which tends to be tedious since it involves close monitoring of the escape region.
In accordance with the present invention, a segmentation control method is provided to sustain chaotic transients in dynamical systems. The sustained transient can be tracked as a system parameter is substantially varied, allowing sustained chaotic transients far away from crisis parameter values.
In accordance with the present invention, a method is provided for controlling the operation of a nonlinear system which is responsive to a parametric signal for sustaining and tracking a chaotic system. The method comprises representing a nonlinear dynamic system as a function of a system parameter and an output value. An initial parametric signal is generated corresponding to a first value of the system parameter. An output signal is produced corresponding to an output value from the nonlinear system in response to the parametric signal. Iterations are performed on the function and parametric perturbations are activated to generate a new parametric signal corresponding to a new value if a current iteration falls within a predetermined neighborhood of a previous iteration.
In accordance with another aspect of the present invention, a controller is provided for the operation of a nonlinear system responsive to parametric signals for sustaining and tracking a chaotic system. The controller comprises a monitor for detecting an output value from the nonlinear system and a processor for calculating iterations of a function representing the nonlinear dynamic system as a function of a parameter value and the output value. The processor is operable to perform parametric perturbations to generate a new parametric value if a current iteration falls within a predetermined neighborhood of a previous iteration. An input device applies the parametric value to the nonlinear system.
In accordance with yet another aspect of the present invention, a system is provided for controlling the operation of a nonlinear system. This system comprises means for generating a parametric signal having a parametric value and controlling means responsive to the parametric signal for controlling the nonlinear system. The controlling means includes a modulator that is responsive to the parametric signal and a feedback signal for producing and applying an input signal to the nonlinear signal to cause the nonlinear system to produce an output signal having an output value which is chaotic. Means responsive to the output signal produce the feedback signal. Correcting means are operable, when a current iteration of a function representing the nonlinear system in terms of the parameter value and the outputted value fall within a predetermined neighborhood of a previous iteration, for performing parameter perturbations to vary the feedback signal.
In accordance with a further aspect of the present invention, a system is provided for tracking the operation of a nonlinear system. This system comprises means for generating a parametric signal at an initial time with an initial selected value and at least one subsequent time with at least one subsequent value different from the initial value. Means responsive to the parametric signal control the nonlinear system at the initial time and at the at least one subsequent time. The controlling means comprises a modulator that is responsive to the parametric signal and a feedback signal for producing and applying an input signal to the nonlinear system to cause the nonlinear system to produce an output signal which is chaotic. Means responsive to the output signal produce and vary the feedback signal when a current iteration of a function representing the nonlinear dynamic system in terms of a value of the parametric signal and a value of the output signal fall within a predetermined neighborhood of a previous iteration.
One key feature of the present invention is that instead of preventing escape to an attractor in advance, the present invention takes a global view of the phase space at the crisis value and eliminates the cause of the crisis. This is achieved through briefly changing the configuration of the phase space by temporarily suppressing a suitable attractor. A main advantage of the present method, not previously found in the art, is that the maintained chaotic state can be tracked (i.e., sustained) over a wide parameter region, whereas other methods involve holding the parameter fixed.
Another key feature of the present invention is that only local knowledge of the dynamics is necessary. In contrast to the prior art algorithms of sustained chaos, knowledge of the dynamics in the region where chaos disappears, or of the pre-iterates of this region are not necessary in the present invention.
Yet another feature of the present invention is the application of algorithms that act by using parameter perturbations.
Another feature of the present invention is an algorithm that allows tracking to sustain chaos. Consequently, chaos is sustained not only at a fixed parametric value. The parametric value can be varied and the algorithm can be applied at the new parametric value.
Another feature of the present invention is the employment of an algorithm that does not require analysis of escape regions which tends to be very tedious and may be impossible in high-dimensional systems.
A further feature of the present invention is an algorithm that can be applied to a dynamic system to sustain chaos even after chaos disappears and the dynamic system has settled on a periodic orbit. Prior art algorithms for sustained chaos fail to provide such a desirable feature.
Still a further feature of the present invention is an algorithm that uses linearization about an attractor rather than about a basin boundary saddle. As a result of this feature, the algorithm is more robust as compared with an algorithm which implements linearization about a basin saddle.
Further features and advantages of the present invention will be set forth in, or apparent from, the detailed description of preferred embodiments thereof which follows.